Denoising of surface EMG with a modified Wiener filtering approach

The correlation dimension D₂ yields good results in several biomedical fields. Nonetheless, no clinical application to electromyography has been developed yet. One reason is the high electromagnetic noise typical of clinical environments. This noise is characterized by sharp spectral lines of variable intensity and frequency. The filtering techniques commonly implemented in electromyographs can efficiently deal with this kind of noise. They allow a safe estimate of linear quantities like the root mean square (r.m.s.) or the median frequency (MF). Their performance is not as good for nonlinear purposes. The filters may modify the nonlinear properties of the signal, leading to unacceptable estimates of D₂. We consider a simple procedure based on a modified Wiener filter. Its performance is compared with that from a bandpass followed by multiple notch filters. Our procedure leads to a gain in precision and accuracy when estimating D₂. The two filtering approaches are also compared with respect to a biomedical application proposed by others. Using data from 12 healthy subjects, the modified Wiener procedure raises the percentage of successes in that application from 17% to 83%. New experimental data suggest D₂ carries information not carried by r.m.s. or MF.     

On the choice of noise for the analysis of the peripheral auditory system

The cross-correlation between output and input of a system containing nonlinearities, when that system is stimulated with Gaussian white noise, is a good estimate of the linear properties of the system. In practice, however, when sequences of pseudonoise are used, great errors may be introduced in the estimate of the linear part depending on the properties of the noise. This consideration assumes special importance in the analysis of the linear properties of the peripheral auditory system, where the rectifying properties of the haircells constitute a second order nonlinearity. To explore this problem, a simple model has been designed, consisting of a second order nonlinearity without memory and sandwiched between two bandpass filters. Different types of pseudonoise are used as input whereupon it is shown that noise based on binary m-sequences, which is commonly used in noise generators, will yield totally incorrect information about this system. Somewhat better results are achieved with other types of noise. By using inverse-repeat sequences the results are greatly improved. Furthermore, certain anomalies obtained in the analysis of responses from single fibers in the auditory nerve are viewed in the light of the present results. The theoretical analysis of these anomalies reveals some information about the organization of the peripheral auditory system. For example, the possibility of the existence of a second bandpass filter in the auditory periphery seems to be excluded.     

Generalized optimal spatial filtering using a kernel approach with application to EEG classification

Common spatial patterns (CSP) has been widely used for finding the linear spatial filters which are able to extract the discriminative brain activities between two different mental tasks. However, the CSP is difficult to capture the nonlinearly clustered structure from the non-stationary EEG signals. To relax the presumption of strictly linear patterns in the CSP, in this paper, a generalized CSP (GCSP) based on generalized singular value decomposition (GSVD) and kernel method is proposed. Our method is able to find the nonlinear spatial filters which are formulated in the feature space defined by a nonlinear mapping through kernel functions. Furthermore, in order to overcome the overfitting problem, the regularized GCSP is developed by adding the regularized parameters. The experimental results demonstrate that our method is an effective nonlinear spatial filtering method.     

Simulation of the effect of local adaption with nonlinear spatial filters

The visual system of vertebrates is capable of processing pattern signals over a wide range of intensity reaching from nearly absolute darkness to very bright sunlight. Typically the visual system of humans extracts fine contours of patterns of sufficiently high intensity or at high background intensity level, showing signal processing properties which can be explained by a bandpass system. Conversely, at very low intensity levels that system shows low-pass response: only coarse contours of patterns are recognized, however, the amplification of the signals has increased. The effect is called local adaption. A model is shown on the basis of a one-stage nonlinear spatial filter which, controlled by the local distribution of pattern intensity, can alter its frequency characteristic between low-pass response and bandpass response. Results are stated for computer-modelled filters. The investigation is restricted to one-dimensional filters, however, the results can be used to explain the function of two-dimensional filters qualitatively.     

Synaptic rectification model equivalent to the correlation-type movement detector

Neural models which are equivalent to the correlation-type movement detector are described. The models involve contrast-coding channels which comprise bandpass linear filters followed by synaptic rectifiers. Linear, one-directional lateral interactions are assumed among the contrast-coding channels. Synaptic rectifiers convert linear spatial interaction into a multiplication-like (quadratic) interaction, which is the core of the correlation-type movement detector. One of the neural models (E-I model) well approximates the correlation model in both time-averaged and dynamic (instantaneous) responses. Possible applicability of the model to movement detection by insects is discussed.     

Intrinsic gain modulation and adaptive neural coding

In many cases, the computation of a neural system can be reduced to a receptive field, or a set of linear filters, and a thresholding function, or gain curve, which determines the firing probability; this is known as a linear/nonlinear model. In some forms of sensory adaptation, these linear filters and gain curve adjust very rapidly to changes in the variance of a randomly varying driving input. An apparently similar but previously unrelated issue is the observation of gain control by background noise in cortical neurons: the slope of the firing rate versus current (f-I) curve changes with the variance of background random input. Here, we show a direct correspondence between these two observations by relating variance-dependent changes in the gain of f-I curves to characteristics of the changing empirical linear/nonlinear model obtained by sampling. In the case that the underlying system is fixed, we derive relationships relating the change of the gain with respect to both mean and variance with the receptive fields derived from reverse correlation on a white noise stimulus. Using two conductance-based model neurons that display distinct gain modulation properties through a simple change in parameters, we show that coding properties of both these models quantitatively satisfy the predicted relationships. Our results describe how both variance-dependent gain modulation and adaptive neural computation result from intrinsic nonlinearity.     

A model of human pattern perception: association fields for adaptive spatial filters.

Visual neurons in the primary visual cortex 'look' at the retinal image through a four-dimensional array of spatial receptive fields (filter-elements): two spatial dimensions and, at each spatial location, two Fourier dimensions of spatial frequency and orientation. In general, visual objects activate filter-elements along each of these dimensions, suggesting a need for some kind of linking mechanism that determines whether two or more filter-elements are responding to the same or different contours or objects. In the spatial domain, a (spatial) association field between filter-elements, arranged to form first-order curves, has been inferred as a flexible method by which different parts of extended (luminance) contours become associated (Field et al., 1993). Linking has also been explored between filters selective for different regions in Fourier space (e.g. Georgeson and Meese, 1997). Perceived structure of stationary plaids suggests that spatial filtering is adaptive: synthetic filters can be created by the linear summation of basis-filters across orientation or spatial frequency in a stimulus-dependent way. For example, a plaid with a pair of sine-wave components at +/-45 deg looks like a blurred checkerboard; a structure that can be understood if features are derived after linear summation of spatial filters at different orientations. However, the addition of an oblique third-harmonic component causes the plaid to perceptually segment into overlapping oblique contours. This result can be understood if filters are summed across spatial frequency, but, in this case, treated independently across orientation. In the present paper, the architecture of an association field is proposed to permit linking and segmentation of filter-elements across spatial frequency and orientation. Three types of link are proposed: (1) A chain of constructive links around sites of common spatial frequency but different orientation, to promote binding of filters across orientation; (2) Constructive links between sites with common orientation but different spatial frequency, to promote binding of filters across spatial frequency; (3) Long-range links between sites of common spatial frequency but different orientation, whose activation and role are determined by activity in a higher spatial frequency band. A model employing the proposed network of links is consistent with at least six previously reported effects on the perception of briefly presented stationary plaids.     

Detection of simple radially symmetric targets: further evidence for the matched filter processing scheme in human pattern detection

The detection of small radially symmetric targets was studied using a subthreshold summation paradigm. Small disc and disc-like patterns with diameters up to 0.6^∘ were used for superposition on Bessel functions of zero order, subthreshold contrast and various spatial frequencies. Contrast interrelation functions prove linear over the whole range of contrasts used for the Bessel functions while their slopes show systematic variation with spatial frequency. An extrapolation of sensitivity from the slopes reveals that sensitivity can be predicted by a simple model assuming detection to be mediated by a transfer function made up as a cascade of an even bandpass function and the disc pattern spectrum, as has been found previously using one dimensional luminance distributions. Problems concerning the formation of pattern-specific radial symmetric filters are discussed. Received: 31 January 2000 / Accepted in revised form: 16 June 2000     

Improved processing of the steady-state evoked potential

Two related procedures for estimating the parameters of steady-state evoked potentials (SSEPs) are introduced. The first procedure involves an initial stage of digital bandpass filtering followed by a Discrete Fourier Transform analysis. In the second method, a high resolution method based on parametric modelling is applied to the filtered data. The digital pre-filter consists of a non-phase shifting Chebychev bandpass filter. The parametric modelling method considers the evoked-response-plus-noise distribution to consist of a set of exponentially damped sinusoids. The frequency, amplitude, phase and damping factors of these components are estimated by calculating the mean of the forward and backward prediction filters and linear regression.We compared the signal-to-noise ratio (SNR) of the new procedures to the conventional Discrete Fourier Transform method for Monte Carlo simulations utilizing known sinusoids buried in white noise, known sinusoids buried in human EEG noise and for a sample of visual evoked potential data. Both of the new methods produce substantially more accurate and less variable estimates of test sinusoid amplitude. For VEP recording, the EEG background noise level is reduced by 5–6 dB over that obtained with the DFT. The new methods also provide approximately 5 dB better SNR than the DFT for detection of sinusoids based on the Rayleigh statistic. The parametric modelling approach is particularly suited for the analysis of very short data records including cycle-by-cycle analysis of the SSEP.     

Characterization of spatial and temporal properties of monkey LGN Y-cells

LGN Y-cells in 3 anaesthetized (N₂O/O₂) and paralyzed rhesus monkeys were investigated with stimuli, intensity modulated by gaussian white noise, and with moving and counterphase modulated spatial sine wave gratings. The results support the model, postulated on the base of electrophysiological recordings in the retina of cat and mudpuppy, which consists of a linear centre and surround mechanism whose responses are modified in a frequency-selective multiplicative way by a nonlinear mechanism in the receptive field. This nonlinear mechanism is also held responsible for the second-order harmonic responses, which are the defining characteristic of Y-cells. The temporal and spatial characteristics of these mechanisms were determined. The responses obtained with the GWN stimulation and with modulated spatial sine wave gratings both indicate that the optimal temporal frequency of the linear mechanisms is near 7 Hz at 70 td and near 5 Hz for the nonlinear mechanism. The optimal spatial frequency for the linear mechanism is between 0.5–2 cycles/deg and between 6–12 cycles/deg for the nonlinear mechanism.     

Firing-rate models for neurons with a broad repertoire of spiking behaviors

Capturing the response behavior of spiking neuron models with rate-based models facilitates the investigation of neuronal networks using powerful methods for rate-based network dynamics. To this end, we investigate the responses of two widely used neuron model types, the Izhikevich and augmented multi-adapative threshold (AMAT) models, to a range of spiking inputs ranging from step responses to natural spike data. We find (i) that linear-nonlinear firing rate models fitted to test data can be used to describe the firing-rate responses of AMAT and Izhikevich spiking neuron models in many cases; (ii) that firing-rate responses are generally too complex to be captured by first-order low-pass filters but require bandpass filters instead; (iii) that linear-nonlinear models capture the response of AMAT models better than of Izhikevich models; (iv) that the wide range of response types evoked by current-injection experiments collapses to few response types when neurons are driven by stationary or sinusoidally modulated Poisson input; and (v) that AMAT and Izhikevich models show different responses to spike input despite identical responses to current injections. Together, these findings suggest that rate-based models of network dynamics may capture a wider range of neuronal response properties by incorporating second-order bandpass filters fitted to responses of spiking model neurons. These models may contribute to bringing rate-based network modeling closer to the reality of biological neuronal networks.     

Nonlinear neuronal mode analysis of action potential encoding in the cockroach tactile spine neuron

 Neuronal mode analysis is a recently developed technique for modelling the behavior of nonlinear systems whose outputs consist of action potentials. The system is modelled as a set of parallel linear filters, or modes, which feed into a multi-input threshold. The characteristics of the principal modes and the multi-input threshold device can be derived from Laguerre function expansions of the computed first- and second-order Volterra kernels when the system is stimulated with a randomly varying input. Neuronal mode analysis was used to model the encoder properties of the cockroach tactile spine neuron, a nonlinear, rapidly adapting, sensory neuron with reliable behavior. The analysis found two principal modes, one rapid and excitatory, the other slower and inhibitory. The two modes have analogies to two of the pathways in a block-structured model of the encoder that was developed from previous physiological investigations of the neuron. These results support the block-structured model and offer a new approach to identifying the components responsible for the nonlinear dynamic properties of this neuronal encoder. Received: 30 December 1994/Accepted in revised form: 25 April 1995     

Detection of aperiodic test patterns by pattern specific detectors revealed by subthreshold summation

The hypothesis that the visual system detects, under certain conditions, stimulus patterns by means of filters matched to these patterns (Hauske et al. 1976) may be challenged by the argument that other coding mechanisms like spatial frequency channels, Gabor or Hermite filters mimick the behaviour of matched filters, a view supported by the finding of non-linear contrast-interrelationship functions (CIFs), as determined in superposition experiments. In this paper we argue that an overall non-linear CIF does not contradict the hypothesis of detection by a single matched filter: we find that the sensitivity functions determined in our experiments can be separated into two components reflecting (i) a bandpass filter and (ii) a filter characterised by the spectrum of the test-pattern. Received: 16 December 1996 / Accepted in revised form: 9 June 1998     

The Essential Complexity of Auditory Receptive Fields

Encoding properties of sensory neurons are commonly modeled using linear finite impulse response (FIR) filters. For the auditory system, the FIR filter is instantiated in the spectro-temporal receptive field (STRF), often in the framework of the generalized linear model. Despite widespread use of the FIR STRF, numerous formulations for linear filters are possible that require many fewer parameters, potentially permitting more efficient and accurate model estimates. To explore these alternative STRF architectures, we recorded single-unit neural activity from auditory cortex of awake ferrets during presentation of natural sound stimuli. We compared performance of > 1000 linear STRF architectures, evaluating their ability to predict neural responses to a novel natural stimulus. Many were able to outperform the FIR filter. Two basic constraints on the architecture lead to the improved performance: (1) factorization of the STRF matrix into a small number of spectral and temporal filters and (2) low-dimensional parameterization of the factorized filters. The best parameterized model was able to outperform the full FIR filter in both primary and secondary auditory cortex, despite requiring fewer than 30 parameters, about 10% of the number required by the FIR filter. After accounting for noise from finite data sampling, these STRFs were able to explain an average of 40% of A1 response variance. The simpler models permitted more straightforward interpretation of sensory tuning properties. They also showed greater benefit from incorporating nonlinear terms, such as short term plasticity, that provide theoretical advances over the linear model. Architectures that minimize parameter count while maintaining maximum predictive power provide insight into the essential degrees of freedom governing auditory cortical function. They also maximize statistical power available for characterizing additional nonlinear properties that limit current auditory models.     

Revealing biogeographical patterns by nonlinear ordinations and derived anisotropic spatial filters

Aim Spatial floristic and faunistic data bases promote the investigation of biogeographical gradients in relation to environmental determinants on regional to continental scales. Our aim was to extract major gradients in the distribution of vascular plant species from a grid‐based inventory (the German FLORKART data base) and relate them to long‐term precipitation and temperature records as well as soil conditions. We present an ordination technique capable of coping with this complex data array. The goal was also to sort out the influence of spatial autocorrelation, assuming floristic autocorrelation is anisotropic. Location Germany, at a spatial resolution of 6′ × 10′. Methods Isometric feature mapping (Isomap) was applied as a nonlinear ordination method. Isomap was coupled to ‘eigenvector‐based filters’ for generating spatial reference models representing spatial autocorrelation. What is novel here is that the derived filters are not based on the assumption of equidirectional autocorrelation. Instead, the so‐called ‘principal coordinates of anisotropic neighbour matrices’ build filters to test the influence of geographical vicinity in directions of high similarity among observations. Results The Isomap ordination of floristic data explained more than 95% of the data variance in six dimensions. The leading two dimensions (representing about 80% of the FLORKART data variance) revealed clear spatial gradients that could be related to independent effects of temperature, precipitation and soil observations. By contrast, the third and higher FLORKART dimensions were dominated by an antagonism of anisotropic spatial autocorrelation and soil conditions. A subsequent cluster analysis of the floristic Isomap coordinates educed the spatial organization of the floristic survey, indicating a considerable sampling bias. Conclusions We showed that Isomap provides a consistent methodical framework for both ordination and derived spatial filters. The technique is useful for tracing the often nonlinear features of species occurrence data to environmental drivers, taking into account anisotropic spatial autocorrelation. We also showed that sampling biases are a conspicuous source of variance in a frequently used floristic data base.     

Generalizations of the Moran effect explaining spatial synchrony in population fluctuations

The Moran effect for populations separated in space states that the autocorrelations in the population fluctuations equal the autocorrelation in environmental noise, assuming the same linear density regulation in all populations. Here we generalize the Moran effect to include also nonlinear density regulation with spatial heterogeneity in local population dynamics as well as in the effects of environmental covariates by deriving a simple expression for the correlation between the sizes of two populations, using diffusion approximation to the theta-logistic model. In general, spatial variation in parameters describing the dynamics reduces population synchrony. We also show that the contribution of a covariate to spatial synchrony depends strongly on spatial heterogeneity in the covariate or in its effect on local dynamics. These analyses show exactly how spatial environmental covariation can synchronize fluctuations of spatially segregated populations with no interchange of individuals even if the dynamics are nonlinear.     

Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex.

Simple cells in the primary visual cortex process incoming visual information with receptive fields localized in space and time, bandpass in spatial and temporal frequency, tuned in orientation, and commonly selective for the direction of movement. It is shown that performing independent component analysis (ICA) on video sequences of natural scenes produces results with qualitatively similar spatio-temporal properties. Whereas the independent components of video resemble moving edges or bars, the independent component filters, i.e. the analogues of receptive fields, resemble moving sinusoids windowed by steady Gaussian envelopes. Contrary to earlier ICA results on static images, which gave only filters at the finest possible spatial scale, the spatio-temporal analysis yields filters at a range of spatial and temporal scales. Filters centred at low spatial frequencies are generally tuned to faster movement than those at high spatial frequencies.     

有色环境噪音对空间异质种群动态同步性的影响

随着种群动态和空间结构研究兴趣的增加,激发了大量的有关空间同步性的理论和实验的研究工作。空间种群的同步波动现象在自然界广泛存在,它的影响和原因引起了很多生态学家的兴趣。Moran定理是一个非常重要的解释。但以往的研究大多假设环境变化为空间相关的白噪音。越来越多的研究表明很多环境变化的时间序列具有正的时间自相关性,也就是说用红噪音来描述更加合理。因此,推广经典的Moran效应来处理空间相关红噪音的情形很有必要。利用线性的二阶自回归过程的种群模型,推导了两种群空间同步性与种群动态异质性和环境变化的时间相关性(即环境噪音的颜色)之间的关系。深入分析了种群异质性和噪音颜色对空间同步性的影响。结果表明种群动态异质性不利于空间同步性,但详细的关系比较复杂。而红色噪音的同步能力体现在两方面:一方面,本身的相关性对同步性有贡献;另一方面,环境变化时间相关性可以通过改变种群密度依赖来影响同步性,但对同步性的影响并无一致性的结论,依赖于种群的平均动态等因素。这些结果对理解同步性的机理、利用同步机理来制定物种保护策略和害虫防治都有重要的意义。     

The impact of stochasticity on the behaviour of nonlinear population models: synchrony and the Moran effect

Environmental variation is ubiquitous, but its effects on nonlinear population dynamics are poorly understood. Using simple (unstructured) nonlinear models we investigate the effects of correlated noise on the dynamics of two otherwise independent populations (the Moran effect), i.e. we focus on noise rather than dispersion or trophic interaction as the cause of population synchrony. We find that below the bifurcation threshold for periodic behaviour (1) synchrony between populations is strongly dependent on the shape of the noise distribution but largely insensitive to which model is studied, (2) there is, in general, a loss of synchrony as the noise is filtered by the model, (3) for specially structured noise distributions this loss can be effectively eliminated over a restricted range of distribution parameter values even though the model might be nonlinear, (4) for unstructured models there is no evidence of correlation enhancement, a mechanism suggested by Moran, but above the bifurcation threshold enhancement is possible for weak noise through phase-locking, (5) rapid desynchronisation occurs as the chaotic regime is approached. To carry out the investigation the stochastic models are (a) reformulated in terms of their joint asymptotic probability distributions and (b) simulated to analyse temporal patterns.